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IOP Conference Series: Materials Science and Engineering
Related content
PAPER • OPEN ACCESS
Constitutive modelling of elastomer/graphene
platelet nanocomposites
To cite this article: Amir A. Abdelsalam et al 2017 IOP Conf. Ser.: Mater. Sci. Eng. 244 012016
- Numerical investigation of large strains of
hyperelastic solids
L R Fakhrutdinov and L U Sultanov
- Elastomeric composites based on carbon
nanomaterials
Sherif Araby, Qingshi Meng, Liqun Zhang
et al.
- Making graphene desirable
View the article online for updates and enhancements.
This content was downloaded from IP address 80.82.77.83 on 25/10/2017 at 10:38
ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
Constitutive modelling of elastomer/graphene platelet
nanocomposites
Amir A. Abdelsalam1, Sherif Araby2,3, M. A. Hassan1,4*and A. A. El-Moneim1,5
1 Materials Science and Engineering Department, Egypt-Japan University of Science
and Technology, Alexandria, Egypt
2 Department of Mechanical Engineering, Faculty of Engineering, Benha University,
Egypt
3 School of Engineering, University of South Australia, SA 5095, Australia
4 Mechanical Engineering Department, Faculty of Engineering Assiut University,
Assiut 71516, Egypt
5 Physical Chemistry Dept., National Research Center, El-Dokki-Giza, Egypt
Email: *mohsen.khozami@ejust.edu.eg
Abstract. Elastomers are used in a wide variety of structural and engineering applications.
They exhibit a nonlinear elastic stress-strain behaviour known as hyperelasticity which is
generally described by hyperelastic strain energy functions. The question raised in the current
study was; which model can accurately describe and predict the actual behaviour of the
elastomer nanocomposites. The tensile data were used to fit the various elastomeric material
models available in MSC.MARC finite element analysis package. The relative percentage error
was calculated to determine the goodness of fit in order to select the best model. Numerical
results showed that the third order deformation model was the best among the various material
models since giving a maximum relative error of fit was 2.7% at small and large strains. To
verify the effectiveness of third order deformation model, FE simulations for tensile test was
carried out. The results showed that the third order model is sufficiently enough to regenerate
the experimental data for uniaxial test and efficiently capture the hyperelastic behavior as good
as the experiments.
1. Introduction
In recent years, rubber products as engineering materials have been used in many industries such as
seals, vehicles tires, hydraulic hoses, and engine mountings. These materials usually exhibit very large
strains, with a strongly non-linear stress–strain relation. The wide range of application of hyperelastic
materials justifies a high research interest in their characterization through experimental techniques
and in the study of accurate constitutive models able to characterize their particular behaviour [1]–[3].
As an important type of polymer material, rubber (also called elastomer) is widely used in industry
due to its reversible and high deformability. The introduction of computers in structural analysis using
finite element analysis (FEA) allows computer simulation of how a part will respond to loads and
forces in its application environment. In other words, it is used to predict the effectiveness of design
concepts of elastomeric components. This is mainly because elastomeric components exhibit large
deformations, nonlinear stress-strain relationships during load, viscoelastic processes such as heat
dissipation and time-dependent behaviour, and nearly incompressible characteristics during
deformation [4]–[7]. A stored energy function can define rubber materials as hyperelastic material.
The coefficients of these functions should be determined by shear test, biaxial, and uniaxial data.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
The biggest problem in characterising the mechanical behaviour of rubber materials is how to
determine the proper strain energy function and material model that provides a good fit with available
experimental data. In this study, different material models groups, such as Mooney-Rivlin, Principle of
stretch and Micromechanical models for incompressible isotropic hyperelastic materials were used to
model and capture the hyper-elastic mechanical properties of SBR with different percentage of
graphene platelets. The relative percentage error was calculated to determine the goodness of data fit
to select the best model that can accurately reproduce the uniaxial tensile test data.
2. Experimental procedures
2.1. Material
The styrene butadiene rubber (SBR, 1502 styrene content 23.5% with a Mooney viscosity of ML
(1+4)) at 100 oC =52). The rubber was reinforced with different graphene platelets (GnPs) to prepare
samples for tensile tests.
2.2. Experimental work
Compounding GnPs into styrene butadiene rubber was done by a laboratory size open two-roll mill
machine at room temperature. The volume ratio of GnPs in composites were 0, 2.4, 5, 10.5 and 16.7
vol.%. The tensile dumb-bell samples of neat SBR and its nanocomposites were die cut to record
stress-strain graphs using Instron 5567 tensile machine with 2 kN load cell. The tensile test was
accomplished according to ASTM D 412 standard [8]. The sample preparations and experimental
procedure details follow a similar process as in Ref. [9].
2.3. Hyperelastic material models for elastomers
The non-linear behaviour of rubber requires specific constitutive material formulations, namely
hyperelastic material models. In most commercial finite element (FE) softwares package such as MSC.
Nastran, MSC. Marc, Abaqus, Ansys, etc., there are different forms of strain energy density functions
W may characterise elastomeric materials. Generally, hyperelastic models are described by strain
energy function density (W) which is formulated as a function depending on different magnitudes
associated with the strain field and the material constants. Usually:
W = W (λ1, λ2, λ3) = W (I1, I2, I3)
(1)
where λ1, λ2, λ3 are principal stretches, and I1, I2, I3 are the invariants of the green deformation tensor.
The strain invariants I1, I2 and I3 in terms of the principal stretch ratios, λ1, λ2, and λ3 are defined as:
(2)
where the stretch ratios are defined as:
(
; i = 1, 2, 3
)
(3)
where εi = ΔLi / Li is the engineering strain. In the case of perfectly incompressible materials, the third
invariant equals one, i.e. I3 = 1.
Three different hyperelastic models were studied to obtain a constitutive material model for elstomer
material, and their formulation are described.
A) Mooney‐Rivlin model [10], [11]
W = C10 (I1  3) + C01 (I2  3)
where C10 and C01 are the material constants
2
(4)
ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
1. Neo‐Hookean model [12]

W=

=
(5)
2. Two term Mooney - Rivlin energy function
W = C10 (I1  3) + C01 (I2  3)
(6)
where C10 and C01 are the material constants
3. Three term Mooney - Rivlin energy function
W = C10 (I1  3) + C01 (I2  3) + C11 (I1  3) (I2  3
(7)
where C10, C01, and C11 are material constants
4. The Signiorini model
W = C10 (I1  3) + C01 (I2  3) + C20 (I1  3)2
(8)
5. Second order invariant
W = C10 (I1  3) + C01 (I2  3) + C11 (I1  3) (I2  3) + C20 (I1  3)2
(9)
6. Third order deformation (or James-Green-Simpson):
W = C10 (I1  3) + C01 (I2  3) + C11 (I1  3) (I2  3) + C20 (I1  3)2 + C30 (I1  3)3
(10)
7. Yeoh model [13]
W = C10 (I1  3) + C20 (I1  3)2 + C30 (I1  3)3
(11)
B) Principle of stretch models
1. Ogden model [14]
∑
)
(

(12)
Usually, the number of terms taken into account in the Ogden models is N = 4. Where μn and αn are
material constants, K is the initial bulk modulus, and J is the volumetric ratio defined by J = λ1λ2λ3.
2. Foam
∑
where
)
(
,
, and

∑
(13)
are material constants.
C) Micromechanical models
1. Arruda‐Boyce model [15]
∑
(
)
*
+
(14)
where Ci and λL are material constants, μ is the initial shear modulus and D is a material constant
related to the bulk modulus.
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
3. Results & discussion
3.1. Data fitting with different rubber material models
Figure 1. (a) – (c) presented, the result of data fitting with different rubber material models used in the
present study. It is clear that the third order deformation model, Figure 1. (b), gave the best curve
fitting if it compared with the other models. Moony3 and second order invariant models are shown in
Figure 1(a) and (b) gave the second best fit after the third order model. The goodness of fit for the
remaining hyperelastic models, Figure 1. (a) –(c) were invariably poor regarding the relative errors of
the least square fitting of uniaxial data with these material models, as it will be explained in Figure 2.
Therefore, the third order deformation model was chosen to represent the hyperelastic behaviour of the
rubber nano-composite materials hereafter in this paper because the relative error does not exceed 3%.
(a)
(b)
(c)
Figure 1. (a)-(c) show the curve fitting of the experimental data with the seven rubber material
models, for pure SBR.
3.2. Effect of the ratio of GnPs % on the values of relative error of fitting
Figure 2 shows the variation of percentage relative error values with GnPs% for various material
models. As shown in Figure 2, the lowest relative error value was observed with third order
deformation model for all GnPs%. Arruda-Boyce model gave the worst fit since the relative error can
reach 7%. The value of relative error in all material models increased with the increasing GnPs %.
This trend was expected to happen because the stress-strain curves showed a marginal increase with
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
GnPs%. Figure 3 illustrates the variation of the material constants C10, C01, C11, C20 and C30 of
third order deformation model with increasing of GnPs %. The material constants, C20 and C30,
slightly affected by GnPs% whereas, C11 and C01 showed a continuous increase with %GnPs
especially after 5% GnPs. On the contrary, C10 showed a decreasing trend with the addition of GnPs.
The reason for this variation may attribute to the mathematical nature of the third order deformation
model which is a polynomial hyperelastic model based on the statistical thermodynamics of crosslinked polymer chains [9-10]. These material constants are empirically determined, and therefore it is
difficult to link directly or indirectly their variations to material anisotropy and inhomogeneity of the
samples.
Figure 2. Percentage of relative error of fit for
the various material models versus GnPs%.
Figure 3. The variation of the material constants
for third order deformation model versus
GnPs %.
3.3. Comparison between FE-simulation and experimental data
3.3.1. The force/displacement curve.
Figure 4 shows a comparison between experiments and FE prediction for load/displacement response
of uniaxial tensile test at different GnPs%. The effect of graphene platelets content, GnPs%, on the
response of the uniaxial tensile test was simulated utilising the material constants obtained from the
third order deformation model. It is seen that even addition of small values of GnPs% affected the
measured stiffness. An increase in the percentage of GnPs% rubber is getting stiffer and the forcedisplacement response was higher. The differences between measured and predicted forcedisplacement response became significant at 16.7% GnPs. However, these differences could be
eliminated if enough data was available and the material constants were derived from the combined
test data. Material constants derived from the synthetic biaxial data provided a better prediction of
rubber behaviour in both tension and simple shear modes [16], [17]. From the engineering point of
view, differences between experimental and FE results, in Figure 4, even at high GnPs% are
acceptable since the maximum relative error did not exceed 7%. The third order deformation model is
sufficiently robust enough to regenerate the experimental data for the uniaxial test.
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
Figure 4. A comparison between experimental and FE- predictions of
load/displacement response of uniaxial tensile test at different GnPs%.
3.3.2. Uniaxial tensile strength.
Figure 5 shows a comparison between experimental and FE- predictions of uniaxial tensile strength at
various GnPs%. In general, the trend of experimental results and FE- simulations are the same up to
10.5 %GnPs after that a slight difference was observed when GnPs% increased. The reason may be
attributed the relative error of fit that increased significantly at GnPs value greater than 10.5 % (see
Figure11). For GnPs% values less than 10%, the relative error of fit for third order deformation model
was below 1% whereas it reached about 3% at 16.7% GnPs. Regardless of differences between
experimental and FE- predictions of tensile strength, still the maximum prediction error did not exceed
4 % even at high GnPs values.
Figure 5. A comparison between experimental and FEpredictions of uniaxial tensile strength at different GnPs%.
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
4. Conclusion
The aim of this paper was to determine the best material model that representing the tensile behaviour
of styrene-butadiene rubber (SBR) at a different volume fraction of graphene platelets (GnPs). To
verify that, different hyperelastic material models have been studied in order to obtain the material
constants of each model. The relative percentage error has been calculated to judge their goodness of
fit and conclude which of them best represents the actual behavior of the rubber nanocomposites. In
view of the obtained results, it concluded that the third order deformation model is the best one to
regenerate tensile tests of SBR /GnPs samples since the relative errors did not research 2.7 %. Biaxial
testing data are needed to minimize the relative error and provide a better prediction of rubber
behaviour especially for rubber nano-composites contains GnPs greater than 10.5 vol. %.
5. References
[1] Ali A, Fouladi M H and Sahari B 2010 A review of constitutive models for rubber-like
materials Am. J. Eng. Appl. Sci. 3 pp.232–239
[2] Sasso M, Palmieri G, Chiappini G and Amodio D 2008 Characterization of hyperelastic
rubber-like materials by biaxial and uniaxial stretching tests based on optical methods Polym.
Test. 27 pp. 995–1004
[3] Hassan M A, Hamdi M and Noma A 2012 The nonlinear elastic and viscoelastic passive
properties of left ventricular papillary muscle of a Guinea pig heart J. Mech. Behav. Biomed.
Mater. 5 pp.99–109
[4] Bhowmick A K 1994 Rubber Products Manufacturing Technology 1st ed. (New York CRC
Press)
[5] Tobajas R, Ibartz E and Gracia L 2016 A comparative study of hyperelastic constitutive models
to characterize the behavior of a polymer used in automotive engines Proc. Int. Electron. Conf.
Mater 2nd. pp.1-17
[6] Martins P A L S, Jorge R M N and Ferreira A J M 2006 A comparative study of several material
models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues
Strain 42 pp.135–147
[7] Nair A U, Lobo H and Bestelmeyer A M 2009 Characterization of damage in hyperelastic
materials using standard test methods and abaqus 2009 SIMULIA Cust. Conf. pp.1–15
[8] ASTM D 412 – 06a 2007 Standard Test Methods for Vulcanized Rubber and Thermoplastic
Elastomers- Tension Annu. B. ASTM Stand. pp.1–14
[9] Araby S, Maged A, Zaman I, Ma J and Majewski P 2013 A facile approach to fabricate
elastomer/graphene platelets nanocomposites Int., Conf., on Smart Materials and
Nanotechnology in Engineering (SMN) 4th, Gold Coast, Australia.
[10] Mooney M 1940 A theory of large elastic deformation J. Appl. Phys. 11 pp.582–592
[11] Rivlin R S and Saunders D W 1951 Large elastic deformations of isotropic materials. vii.
experiments on the deformation of rubber Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 243
pp.251–288
[12] Treloar L R G 1943 The elasticity of a network of long‐chain molecules-II Trans. Faraday Soc.
39 pp.241–246
[13] Yeoh O H 1990 Characterization of elastic properties of carbon-black-filled rubber vulcanizates
Rubber Chemistry and Technology 63 pp.792–805
[14] Ogden B Y R W 1972 Large deformation isotropic elasticity - on the correlation of theory and
experiment for incompressible rubberlike solids Proc. R. Soc. A 326 pp.565–584
[15] Arruda E M and Boyce M C 1993 A three-dimensional constitutive model for the large stretch
behavior of rubber elastic materials J. Mech. Phys. Solids 41 pp.389–412
[16] Wangkiet R, Sucharitpwatskul S, Sirisinha C, Deeprasertkul C, and Sae-Oui P 2008
Characterization of material constants based on synthetic biaxial data ScienceAsia, 34 pp.395–
399
[17] Hassan M A, Abouel-kasem A, El-sharief M A and Yusof F 2012 Evaluation of the material
constants of nitrile butadiene rubbers (NBRs) with different carbon black loading (CB): FEsimulation and experimental Polymer 53 pp.3807–3814
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ICMIM 2017
IOP Publishing
IOP Conf. Series: Materials Science and Engineering
244 (2017) 012016 doi:10.1088/1757-899X/244/1/012016
1234567890
Acknowledgements
The authors would like to thank the Ministry of Higher Education (MOHE) in Egypt for their financial
support for achieving the current research. Also, they would like to thank the Japan International
Cooperation Agency (JICA) for their continuous support to Egypt-Japan University of Science and
Technology (E-JUST).
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